Present Value

Present Value Calculations in Health Economic Evaluation

Present value expresses a cost or a health effect that occurs in a later year as an equivalent amount at a single reference date, usually the start of the analysis. Every discounted total in a cost-effectiveness results table is a present value, whether it is measured in pounds or in QALYs, and the NICE manual (PMG36, section 4.5.1) states that cost-effectiveness results should reflect the present value of the stream of costs and benefits over the time horizon. The arithmetic for one amount is short, but the reference date, the timing of each amount within a year or model cycle, the price basis and the rate schedule all change the answer. The reasons for discounting and the choice of rate are treated on the discounting and discount rate pages, and present value is the calculation that applies them.

Bringing one amount or a stream to a reference date

A present-value calculation weights each future amount by a discount factor that falls as the delay grows. Attema, Brouwer and Claxton illustrate the idea with a bond paying 2% a year, under which 102 euros next year has a present value of 100 euros today and 100 euros next year is worth 98.04 euros now. They give the discount factor for a period as one divided by one plus the rate, raised to the power of the number of periods, and note that this constant model is the one most often used by social planners.

$$PV = \sum_{t=0}^{T} \frac{X_t}{(1+r)^t} = \sum_{t=0}^{T} d_t X_t$$

where $PV$ is the present value at the reference date, $X_t$ is the cost or health effect occurring $t$ years after that date, $r$ is the annual discount rate written as a decimal, $T$ is the last year of the time horizon and $d_t = (1+r)^{-t}$ is the discount factor for year $t$. An amount at $t = 0$ has a factor of 1 and is not reduced.

Running the same formula forwards gives the future value, $X_t = PV (1+r)^t$, which compounds a present sum to a later date. Economic evaluation works almost entirely backwards, because the decision is taken now, and the time value of money page gives the general principle behind both directions.

The formula is indifferent to units. A stream of yearly QALYs has a present value in QALYs, and a stream of costs has a present value in pounds, so a cost-utility result divides one present value by another and never adds them. Where benefits are valued in money, as in cost-benefit analysis, the present value of benefits less the present value of costs gives the net present value, and their ratio gives the benefit-cost ratio.

Closed forms for level, growing and perpetual streams

Many streams in health care models are level or grow at a steady rate, such as a fixed yearly drug cost or a constant yearly QALY gain. Summing such a stream year by year and using a closed form give the same answer, and the closed form is the quicker check on a spreadsheet.

$$A(r,n) = \sum_{t=1}^{n} \frac{1}{(1+r)^t} = \frac{1 - (1+r)^{-n}}{r}$$

where $A(r,n)$ is the annuity factor, the present value of 1 paid at the end of each of $n$ years at annual rate $r$. A level amount $X$ a year then has a present value of $X \times A(r,n)$. When each amount falls at the start of the year instead, every amount is discounted for one period less and the present value is $X \times A(r,n) \times (1+r)$.

$$PV_g = \frac{X_1}{r-g} \left[ 1 - \left( \frac{1+g}{1+r} \right)^{n} \right]$$

where $PV_g$ is the present value of a stream that starts at $X_1$ at the end of year 1 and grows by a constant proportion $g$ each year for $n$ years, $r$ is the discount rate and $r$ differs from $g$. As $n$ grows without limit and $r$ exceeds $g$, the bracket tends to 1 and the stream becomes a growing perpetuity worth $X_1 / (r-g)$, or $X / r$ when $g$ is zero. At 3.5% an illustrative GBP 1,000 a year for ever is worth 1,000 / 0.035 = GBP 28,571, which shows that even an endless stream has a finite present value.

The annuity factor also runs in reverse. Annualisation divides a capital outlay by $A(r,n)$ to give an equal yearly charge over an asset's useful life, and annuity payment schemes for one-off therapies use the same factor to set instalments.

Timing within a year or a model cycle

The formula assumes each amount falls at a known point in time, and the convention chosen moves every factor. Counting a year's costs at its end, at its start or at its midpoint gives three different present values, and the midpoint convention multiplies each end-of-year factor by $(1+r)^{0.5}$, which is 1.0173 at 3.5%. The half-cycle correction in a Markov model addresses a related question about when within a cycle transitions occur, and the two adjustments need to be set consistently.

When the cycle length is shorter than a year, the annual rate has to be converted to a per-cycle rate that compounds back to the annual rate.

$$r_c = (1+r)^{1/k} - 1$$

where $r_c$ is the discount rate per cycle, $r$ is the annual rate and $k$ is the number of cycles in a year. Dividing the annual rate by $k$ instead overstates the discount, because the per-cycle rates then compound to more than $r$ over the year.

Models evaluated in continuous time, for example a partitioned survival model whose survival curves are integrated directly instead of summed by cycle, use the factor $e^{-\rho t}$ in place of $(1+r)^{-t}$.

$$\rho = \ln(1+r)$$

where $\rho$ is the continuously compounded rate that gives the same factor at each whole year as the annual rate $r$. At 3.5%, $\rho$ is 0.0344, and $e^{-0.0344 \times 10}$ equals $(1.035)^{-10}$, or 0.7089.

Rate schedules and the price basis

A present value is only as consistent as the rate and the values fed into it. The HM Treasury Green Book states that its discount rate is a real rate that must be applied only to values from which general inflation has already been removed, and that adding the inflation rate and the discount rate together produces a mathematically incorrect result. The exact link between the two is multiplicative, one plus the nominal rate being the product of one plus the real rate and one plus inflation, and the Green Book avoids the problem by converting values to real terms first and then discounting them.

The NICE manual (PMG36) uses one constant rate. Section 4.5.1 sets 3.5% a year for both costs and health effects in the reference case, and section 4.5.3 allows the committee to consider 1.5% for both only when its stated criteria are all met. The discount rate page sets out those criteria and the rates in other guidelines.

The Green Book uses a declining schedule. Its supplementary guidance on discounting, updated in October 2026, sets 3.00% for years 1 to 30, 2.50% for years 31 to 75 and 2.25% for years 76 to 125, and states that these are forward rates, not spot rates. For a horizon ending in year 40, the 3.00% rate applies to the first 30 years and 2.50% only to years 31 to 40, so the factor for year 40 is a product of the two.

$$d_{40} = 1.03^{-30} \times 1.025^{-10} = 0.41199 \times 0.78120 = 0.3218$$

where $d_{40}$ is the discount factor for year 40 under the schedule, matching the 0.3218 in the guidance's own table. An illustrative GBP 1,000 in year 40 is therefore worth about GBP 322 today. Applying 2.50% to all 40 years would give a factor of 0.3724, or about GBP 372, overstating the present value by roughly 16%.

The Green Book now handles the growing value of health through the values themselves. The February 2026 edition discounted health and life effects at a lower rate of 1.5%, while the edition updated on 28 October 2026 has no separate health rate and states that the value of future real health effects should be uprated by 2% a year, with the willingness-to-pay value for a QALY given as GBP 70,000 in 2020-21 prices. Attema and colleagues note this route as an alternative to differential discounting, inflating QALY estimates to reflect their increased value. With uprating at 2% and discounting at 3.00% in years 1 to 30, each year's factor is $(1.02/1.03)^t$, which equals constant discounting at 1.03 divided by 1.02, minus 1, or about 0.98% a year.

Worked example with a screening programme over ten years

All numbers in this example are illustrative. A community screening programme needs GBP 600,000 of equipment at the start (year 0) and GBP 120,000 of running costs in each of the next 10 years, and it adds 12 QALYs a year over the same 10 years compared with no programme. Running costs and QALYs are counted at the end of each year and discounted at the NICE reference-case rate of 3.5%.

1. Discount factors. The factor for year 1 is 1 / 1.035 = 0.96618 and for year 10 is $(1.035)^{-10}$, or 0.70892. The sum of the ten factors is the annuity factor A(0.035, 10), equal to 1 minus 0.70892, divided by 0.035, or 8.31661.

2. Present value of costs. Running costs are worth 120,000 × 8.31661 = GBP 997,993. The equipment is already at the reference date, so the total is 600,000 + 997,993 = GBP 1,597,993, against GBP 1,800,000 undiscounted.

3. Present value of QALYs. The gain is 12 × 8.31661 = 99.80 discounted QALYs, against 120 undiscounted.

4. Cost per QALY gained. The discounted ratio is 1,597,993 / 99.80 = GBP 16,012 per QALY, against 1,800,000 / 120 = GBP 15,000 undiscounted. Discounting raises the ratio because the equipment cost is not discounted while every QALY is.

5. Midpoint timing. Counting running costs and QALYs in the middle of each year multiplies the annuity factor by 1.035 to the power 0.5, or 1.01735, giving 8.46089. Costs become 600,000 + 120,000 × 8.46089 = GBP 1,615,307, QALYs become 12 × 8.46089 = 101.53, and the ratio falls slightly to GBP 15,910.

6. Monthly cycles. If the same model ran in monthly cycles, the correct rate per cycle is $(1.035)^{1/12} - 1$, or 0.2871%, and the factor after 120 cycles is 0.70892, the same as the annual factor for year 10. Dividing 3.5% by 12 instead gives 0.2917% a cycle and a factor of 0.70505 after 120 cycles, because that rate compounds to 3.557% a year.

7. Spreadsheet check. In Excel, =PV(3.5%,10,-120000) returns 997,993 for the end-of-year running costs, and adding 1 as the type argument returns 1,032,922, which is 997,993 × 1.035 for payments at the start of each year.

The discounted cost per QALY of about GBP 16,000 is the figure a committee would compare with its threshold, and the undiscounted figures and the timing choice belong in the report beside it. A programme with this profile looks slightly less attractive after discounting, and the gap would widen with a longer delay between the outlay and the gains.

Reading present values in appraisal

Present values are the inputs to the summary measures, not the measures themselves. An ICER divides the difference in present value of costs between two options by the difference in present value of QALYs, and net monetary benefit converts the QALY difference to money before subtracting. Because a single rate and timing convention sit inside every total, PMG36 (section 4.10.4) asks for the disaggregated table of costs, QALYs and life years to be presented with and without discounting, and a sensitivity analysis over the rate shows how far the conclusion depends on it.

Two other uses of the phrase need to be kept apart. PMG36 (section 4.4.12) asks that older costs be adjusted to present value using inflation indices such as the NHS cost inflation index, which is a change of price year and not discounting. The Green Book expresses public sector financial impacts in nominal terms and not in present value terms, so a budget impact analysis and a present-value appraisal of the same programme answer different questions.

Common errors in present-value calculations

Most mistakes come from a mismatch between the formula and the timing, units or rate it is given. Four of them are specific to present-value arithmetic, and the general discounting errors are listed on the discount rate page.

• Dividing the annual rate by the number of cycles. The per-cycle rate is $(1+r)^{1/k} - 1$, as step 6 shows.
• Reading a declining schedule as spot rates. The Green Book rates apply to the years in their band only, so later factors are products of earlier ones.
• Discounting past costs forward. The Green Book states that discounting should not be applied retrospectively to costs and benefits that have already occurred.
• Forgetting the sign convention. Excel's annuity functions treat cash paid out as negative, so =PV with a positive payment returns a negative present value.

Sources

• Attema AE, Brouwer WBF, Claxton K. Discounting in economic evaluations. PharmacoEconomics. 2018;36(7):745-758. doi:10.1007/s40273-018-0672-z
• HM Treasury. The Green Book (2026). London: HM Treasury; updated 28 October 2026. Chapters 6 (Discounting), 8 (Life and health) and 9 (Presentation of results). https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government/the-green-book-2026
• HM Treasury. The Green Book (2026), edition updated 5 February 2026, chapter 6, health discount rate (superseded).
• HM Treasury. Green Book supplementary guidance: discounting. London: HM Treasury; October 2026. Chapter 3 and Annex A. https://www.gov.uk/government/publications/green-book-supplementary-guidance-discounting
• Microsoft. PV function. Microsoft Support; accessed 28 October 2026. https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). Chapter 4, economic evaluation, sections 4.4.12, 4.5 and 4.10.4. Published 31 January 2022, last updated 13 October 2026. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation